Combinatorial algorithms for some multiﬂow...

Combinatorial algorithms for some multiflow problems

and related network designs

Hiroshi Hirai The University of Tokyo

Connectivity Workshop HIM, Bonn, September 7-11, 2015

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Our result First combinatorial polytime algo for I First combinatorial strongly-polytime algo for II

Feature Build on Discrete Convex Analysis beyond

Application Approx. of terminal backup & node-multiflow cut

I. Mincost node-demand multiflow problem II. Maximum node-capacitated multiflow problem

We address

Zn

2

I. Mincost node-demand multiflow problem

N = (V,E,c,a,T): undirected networkc: E → Z+: edge-capacity a: E → Z+: edge-cost T: terminal set (⊂ V) r: T → Z+: demand

Def: Multiflow <=> f: { T-paths } → R+ s.t. f(e) := ∑ { f(P) | P: e ∈ P} ≤ c(e) (e ∈ E)

T

3

feasible <=> (s,T-s)-flow in f ≥ r(s) (∀s in T)

I. Mincost node-demand multiflow problem

N = (V,E,c,a,T): undirected networkc: E → Z+: edge-capacity a: E → Z+: edge-cost T: terminal set (⊂ V) r: T → Z+: demand

Def: Multiflow <=> f: { T-paths } → R+ s.t. f(e) := ∑ { f(P) | P: e ∈ P} ≤ c(e) (e ∈ E)

T

3

feasible <=> (s,T-s)-flow in f ≥ r(s) (∀s in T)

Find a feasible multiflow f of minimum total cost ∑ a(e)f(e)

Introduced by Fukunaga (2014) as LP-relaxation of a class of network designs

s ≥r(s)

Min. ∑ a(e) f(e) s.t. f: multiflow (s,T-s)-flow in f ≥ r(s) (s ∈ T)

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Min. ∑ a(e) x(e) s.t. ∑ {x(e) | e ∈ ∂X} ≥ r(s) (s ∈ T, (s,T-s)-cut X) 0 ≤ x(e) ≤ c(e) (e ∈ E)

x(e) = f(e)s ≥r(s)


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Min. ∑ a(e) x(e) s.t. ∑ {x(e) | e ∈ ∂X} ≥ r(s) (s ∈ T, (s,T-s)-cut X) 0 ≤ x(e) ≤ c(e) (e ∈ E)

x(e) = f(e)Lovasz-Cherkasskys ≥r(s)


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Min. ∑ a(e) x(e) s.t. ∑ {x(e) | e ∈ ∂X} ≥ r(s) (s ∈ T, (s,T-s)-cut X) x(e) in {0,1,2,…,c(e)} (e ∈ E)

Terminal backup problem

Anshelevich-Karagiozova 11: r = 1 (> c = 1) P Bernath-Kobayashi-Matsuoka 13: c = ∞ P

P or NP-hard ??

very special class of skew-supermodular covering

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Min. ∑ a(e) x(e) s.t. ∑ {x(e) | e ∈ ∂X} ≥ r(s) (s ∈ T, (s,T-s)-cut X) x(e) in {0,1,2,…,c(e)} (e ∈ E)

Terminal backup problem

Anshelevich-Karagiozova 11: r = 1 (> c = 1) P Bernath-Kobayashi-Matsuoka 13: c = ∞ P

naive

Fukunaga 14: half-integrality of LP-relax. (= Multiflow-relax.) half-integral opt. > 2-approx. > 4/3-approx.obtained by

ellipsoid method

P or NP-hard ??

very special class of skew-supermodular covering

5

n = |V|, m = |E|, k = |T|, A= max a(e), C =∑ c(e)

O(n log (n AC) MF(kn, km)) time algorithm to find half-integral opt. of


> combinatorial implementation of 4/3-approx. of terminal backup

Result

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Our multiflow problem generalizes mincost maximum free multiflow problem (Karzanov 79)

Goldberg, Karzanov 97: combinatorial weakly polynomial time algorithms … O(??)

Karzanov 94: strongly polynomial time algorithm … ellipsoid

Open problem: combinatorial strongly polynomial time algorithm ?

Remark

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Idea Discrete Convex Analysis (Murota) ~ theory of “convex” optimization over Zn

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Idea Discrete Convex Analysis (Murota) ~ theory of “convex” optimization over

Adapting DCA algorithm & technique (steepest descent algo. & proximity scaling)

Zn

8

Dual of our multiflow problem ~ “convex” optimization over

n

123

0≃

n

LP-dual (~~ cut packing)

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characteristic vector

�(X) � 0 (s � T, X : (s, T \ s)-cut)

�

s�T, X:(s,T\s)-cut

r(s)�(X) ��

ij�E

c(ij) max{0,�

X

�(X)[�X](ij) � a(ij)}Max.

s.t.

st

u

v

�

s�T

r(s)D(xs, O) ��

ij�E

c(ij) max{0, D(xi, xj) � a(ij)}Max.

s.t.

(x1, x2, . . . , xn) �

n

O

xy

D(x,y)

in s-th branchxs

( s in T )

10

V={1,2,…,n}

�

s�T

r(s)D(xs, O) ��

ij�E


s.t.

(x1, x2, . . . , xn) �

n

O

xy

D(x,y)

in s-th branchxs

( s in T )

half-integrality10

1/2

V={1,2,…,n}

�

s�T

r(s)D(xs, O) ��

ij�E


s.t.

(x1, x2, . . . , xn) �

n

O

xy

D(x,y)

in s-th branchxs

( s in T )

half-integrality

“discretely concave”

10

1/2

V={1,2,…,n}

�L -convex function: (Murota, Murota-Fujishige, Favati-Tardella)

•Optimality check > Submodular Func. Min. •Steepest Descent Algorithm by successive SFMs • -bound of # iterations of SDA (Kolmogorov-Shioura 04) •Proximity & domain scaling techniquel�

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g(x) + g(y) � g(

�x + y

2

�) + g(

�x + y

2

�) (x, y � Zn)

x

y

�x + y

2

�

�x + y

2

�

(negative of) our dual objective satisfies

•Optimality check > k-Submodular Func. Min. •Steepest Descent Algorithm by successive k-SFMs • -bound of # iterations of SDA •Proximity & domain scaling techniquel�

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g(x) + g(y) � g(

�x + y

2

�) + g(

�x + y

2

�) (x, y � Zn)==

n

(Huber, Kolmogorov 12)

x

y

•k-SFM in oracle model ~~ P or NP-hard ?? •k-SFM in VCSP model ~~ P (Thapper, Zivny 12) •Our special case ~~ O(MF(kn,km))

SDA + scaling > dual opt x* in O(n log (n AC) MF(kn, km)) time

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(Iwata,Wahlstrom,Yoshida 14)

n

min g(x) s.t. x ∈

•k-SFM in oracle model ~~ P or NP-hard ?? •k-SFM in VCSP model ~~ P (Thapper, Zivny 12) •Our special case ~~ O(MF(kn,km))

SDA + scaling > dual opt x* in O(n log (n AC) MF(kn, km)) time

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x* > primal opt f* by complementary slackness with x*

one circulation problem

(Iwata,Wahlstrom,Yoshida 14)

n

min g(x) s.t. x ∈

1.5

1.53

2

2

12

53

4

9

T

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II. Maximum node-capacitated free multiflow problem

c:V \ T > Z+

N = (V,E,c,T)

Max. ∑ f(P) s.t. f: multiflow

•dual of LP-relaxation of node-multiway cut •dual half-integrality > 2-approximation

Garg, Vazirani, Yannakakis 04

Pap 07,08 •primal half-integrality •strong polynomial time solvability (ellipsoid)

Babenko, Karzanov 08 •combinatorial O(MF(n, m, C) n^2 (log n)^2 log C) time

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Mader’s openly-disjoint T-path thm

ResultO(m (log k) MSF(n, m,1)) time algorithm to find half-integral primal & dual opt.

MSF(n,m,h): max. submodular flow on network of n nodes, m edges, time complexity h of computing exchange capacity

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ResultO(m (log k) MSF(n, m,1)) time algorithm to find half-integral primal & dual opt.

MSF(n,m,h): max. submodular flow on network of n nodes, m edges, time complexity h of computing exchange capacity

Fujishige, Zhang 92: O(n^3 h) > O(m n^3 log k) > 2-approx. of node-multiway cut in O(m n^3 log k) time

c.f. Madan-Chekuri 15: (2+ε)-approx. in O(mn/ε^2) time ~

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SketchDual objective is again “L-convex” on

0

1/2

1

3/2

2

�1/2

�1

�3/2

�2

n

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0

1/2

1

3/2

2

�1/2

�1

�3/2

�2

n•Steepest descent algo. • -bound • Steepest direction < Max. subflow • Dual opt > primal opt subflow feasibility

l�

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0

1/2

1

3/2

2

�1/2

�1

�3/2

�2

n•Steepest descent algo. • -bound • Steepest direction < Max. subflow • Dual opt > primal opt subflow feasibility

l�

bisubflow>subflow

perturbation

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• H. Hirai: L-extendable functions and a proximity scaling algorithm for minimum cost multiflow problem, Discrete Optimization, 18 (2015), 1-37.

• H. Hirai: A dual descent algorithm for node-capacitated multiflow problems and its applications, 2015, arXiv:1508.07065.

Thank you for your attention !

http://www.misojiro.t.u-tokyo.ac.jp/~hirai/

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